Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as sgn.

Definition

The signum function of a real number x is defined as follows:

Properties

Any real number can be expressed as the product of its absolute value and its sign function:

It follows that whenever x is not equal to 0 we have

Similarly, for any real number x,

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero): Note, the resultant power of x is 0, similar to the ordinary derivative of x. The numbers cancel and all we are left with is the sign of x.

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function, which can be demonstrated using the identity

Ho Chi Minh

Hồ Chí Minh (/ˈhoʊ ˈtʃiː ˈmɪn/; Northern Vietnamese pronunciation: [ho̞˧˩ t͡ɕi˧˥ mɪŋ˧], Southern Vietnamese pronunciation: [ho̞˧˩ t͡ɕɪj˧ mɪ̈n˧]; 19 May 1890 – 2 September 1969; Chữ nôm: 胡志明), born Nguyễn Sinh Cung, also known as Nguyễn Tất Thành and Nguyễn Ái Quốc, was a Vietnamese Communist revolutionary leader who was prime minister (1945–55) and president (1945–69) of the Democratic Republic of Vietnam (North Vietnam). He was a key figure in the foundation of the Democratic Republic of Vietnam in 1945, as well as the People's Army of Vietnam (PAVN) and the Việt Cộng (NLF or VC) during the Vietnam War.

He led the Việt Minh independence movement from 1941 onward, establishing the Communist-ruled Democratic Republic of Vietnam in 1945 and defeating the French Union in 1954 at the battle of Điện Biên Phủ. He officially stepped down from power in 1965 due to health problems, but remained a highly visible figurehead and inspiration for those Vietnamese fighting for his cause—a united, communist Vietnam—until his death. After the war, Saigon, the former capital of the Republic of Vietnam, was renamed Hồ Chí Minh City.